with initial condition
\(\mathbf {x}[0] = \mathbf {x}_{0} \in \mathbb {X}_{0} \lbrace \mathbf {x}_{0} \in \mathbb {R}^{n} \mid \text{problem (\ref {eq:mpc}) is feasible}\rbrace \) 
is the set of all admissible initial states.
For well-posedness of MPC, we require \( \mathbb {X} \subseteq \mathbb {X}_{0}\) .
Let
\(J(\mathbf {x}[i]) \sum _{k=0}^{N-1}{c(\mathbf {x}[k|i], \mathbf {u}[k|i])} + c_{f}(\mathbf {x}[N|i])\) 
be the cumulative cost and \(J^{*}(\mathbf {x}[i]) \sum _{k=0}^{N-1}{c(\mathbf {x}^{*}[k|i], \mathbf {u}^{*}[k|i])} + c_{f}(\mathbf {x}^{*}[N|i])\)  be the optimal cumulative cost.
By Lyapunov stability theorem, we can derive the following sufficient condition for asymptotic stability of MPC as in Theorem 12.2 in [1]}.
